Sequences and Limits
In this handout, we develop a precise definition of what it means for a sequence to converge to a number and use it to distinguish between a sequence approaching a value and a sequence actually attaining that value.
1. Two Infinite Lists of Numbers
You are given two infinite lists of numbers. You are not told their formulas — only their first few terms.
| Sequence A | Sequence B | ||
|---|---|---|---|
| $n$ | $a_n$ | $n$ | $b_n$ |
| 1 | 0.50 | 1 | 1 |
| 2 | 0.67 | 2 | $-1$ |
| 3 | 0.75 | 3 | 1 |
| 4 | 0.80 | 4 | $-1$ |
| 5 | 0.83 | 5 | 1 |
| 10 | 0.91 | 6 | $-1$ |
| 50 | 0.98 | $\vdots$ | $\vdots$ |
| $\vdots$ | $\vdots$ | ||
Question: What do you notice?
2. Walking Toward a Wall
Imagine you're walking toward a wall. You start 10 meters away from a wall. At each step, you walk half of the remaining distance to the wall.
Two Questions:
- After many steps, is your distance from the wall getting smaller?
- Will you ever reach the wall in a finite number of steps?
Definition
A sequence is a function $f \colon \mathbb{N} \to \mathbb{R}$. We write $f(n)=a_n$ and denote the sequence by .
Examples: Common Ways to Describe Sequences
- Explicit list: $\left(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\right)$
- Formula: $(a_n)_{n=1}^{\infty}$ where $a_n = $
- Recursive: $(x_n)$ where $x_1 = 2$ and $x_{n+1} = \frac{1}{2}(x_n + 1)$ for $n \ge 1$
- General term: $((-1)^{n})_{n=1}^{\infty} = (-1,1,-1,1,-1,1, \ldots)$
Note: A sequence is an list, not a set.
Informally, a sequence $(a_n)$ converges to $a$ if: after some point, all terms of the sequence stay close to $a$.
Question: Can you explain why Sequence A "feels" convergent and Sequence B does not?
We need a precise definition:
Definition: Convergence of a Sequence
A sequence $(a_n)$ converges to $a \in \mathbb{R}$ if, for every $\varepsilon >0$, there exists $N \in \mathbb{N}$ such that whenever , it follows that
$$$$In this case, we write
$$\lim_{n \to \infty} a_n=a, \quad \text{or} \;\; a_n \to a.$$Definition: $\varepsilon$-Neighborhood
Given $a \in \mathbb{R}$ and $\varepsilon > 0$, the $\varepsilon$-neighborhood of $a$ is
$$V_\varepsilon(a) = = $$Geometric Interpretation
$(a_n) \to a$ means: "Given any $\varepsilon$-neighborhood of $a$, eventually terms of $(a_n)$ are in that neighborhood."
Example: Prove $\lim\limits_{n \to \infty} \frac{1}{\sqrt{n}} = 0$
Scratch Work (DO THIS FIRST!):
We want: .
Solving for $n$:
Formal Proof:
Let $\varepsilon > 0$ be arbitrary.
Choose $N \in \mathbb{N}$ such that .
Now let $n \geq N$. Then:
Therefore, $\lim\limits_{n \to \infty} \frac{1}{\sqrt{n}} = $ .
Template for Convergence Proofs
To prove $\lim\limits_{n \to \infty} a_n = a$:
- Start: Let $\varepsilon>0$.
- Scratch work: Solve $|a_n-a|<\varepsilon$ for $n$.
- Choose $N$: Choose $N$ accordingly.
- Verify: For $n\ge N$, verify $|a_n-a|<\varepsilon$.
- Conclude: $\lim\limits_{n \to \infty} a_n=a$.
Exercise 1
Prove that $\lim\limits_{n \to \infty} \frac{n+1}{n} = 1$ using the definition of convergence of a sequence.
Definition
A sequence that converge is said to be a divergent sequence.
How to Prove a Sequence Diverges
Remember: $a_n \to a$ if for every $\varepsilon >0$, there exists $N \in \mathbb{N}$ such that whenever , it follows that $|a_n-a|< \varepsilon$.
Negation of convergence:
$(a_n) \not\to a$ means: $\varepsilon > 0$ such that $N \in \mathbb{N}$, $n \geq N$ such that .
Example: Show that $(-1, 1, -1, 1, -1, 1, \ldots)$ Does Not Converge to 0
Exercise
Prove that $$\lim_{n \to \infty} \left(\frac{2n+1}{5n+4}\right) = \frac{2}{5}$$ using the definition of convergence of a sequence.
← Back to MATH 4630